 Okay, so back to English, thanks again everybody for being here, attending the mini course today. It will be my freedom day because it's my last talk and I expect to speak at most for one hour and 15 minutes, okay, depending on how we proceed, but it will not be like the two hours that I did in all the previous lectures. And why do I intend to do that? Because I will try not to focus on any fancy machinery of the proofs and just a kind of a blueprint of how we make the applications on the existence of Markov partitions. As a matter of fact, the goal of today is exactly to present these applications whenever we know the existence of Markov partitions and certainly in the case of non-uniformly hyperbolic systems, but not only this, but also some uniformly hyperbolic systems like the billiard given by the law and SCAS, it's a kind of uniformly hyperbolic system, but because it has singularities it introduces a new difficulty that was not able to be understood before and it's only able to be understood using these techniques that we have discussed in the previous four lectures that were developed for non-uniformly hyperbolic systems and in particular also work for these uniformly hyperbolic systems with singularities. And the overall idea for the applications is why having a Markov partition provides you these applications. Well, recall that whenever you have this Markov partition with all the particularities, the specifications that we have required for it, we can find an extension by a topological Markov shift. So the idea of applications is to understand some dynamical properties of the geometrical dynamical system that you have and whenever this dynamical property that you are interested in lives inside this non-uniformly hyperbolic set with chi and sharp, then you can try to understand them by understanding them at the level of the symbolic dynamics. So whenever it is easier to understand this property at that symbolic level, then we can get the original object that we want to understand, lift it either to sigma or to sigma sharp, understand it at the symbolic level and then project back to the original geometrical model. So in summary, so far the applications for the existence of this Markov partitions in the non-uniformly hyperbolic context, I can summarize in these four different topics. The first one is understanding measures of maximal entropy and inside this class we can understand how many measures of maximal entropy we have. In some situations we can also understand when this measure of maximal entropy is unique and going further and here I'm mentioning a very, very recent result, you can also get exponential decay of correlations. So now it is all possible to understand these three properties at least for surface defilmophisms. In general, in higher dimensions we can only understand, we can only bound in some sense the number of measures of maximal entropy, but these two questions are still open if you go to higher dimensions. Another application that we can that we can get is establishing ergodic properties of measures of these measures of maximal entropy and more generally of equilibrium states associated to hold so the ultimate goal in this context at least from the measure theoretical point of view is to establish the Bernoulli property is to say that this measure from the measure theoretical point of view is nothing but a kind of coin tossing or a full shift with a Bernoulli measure. So this is also possible whenever you have the presence of these Markov partitions. The third application is counting periodic trajectories, either periodic points or closed orbits of flows. And more recently there has been also a development due to Snebenovadia in which he uses the approach, the original approach of Sinai to construct SRB measures. So if you recall the original approach of Sinai to do it was exactly using Markov partitions. So he got these Markov partitions, he understood some extra properties of these Markov partitions in particular where the coding map that comes from Sigma to M is injective. And then using these injectivity properties he was able to construct to redo with many extra difficulties that come into play during the proofs redo this Sinai approach to construct these like conditional measures on the symbolic level and then project them back to the manifold and get these hyperbolic SRB measures. Actually what he gets is a sufficient and necessary condition for the existence of such measures. Okay so let's focus first in the first topic which is of measures of maximal entropy and let me for easy of understanding at least in the first results fix myself on the usual setting that we were explaining all ideas. So what is the usual setting? You have a surface, so it's a map F defined on a closed surface. You assume that this map is C1 plus beta, it is a different morphism. And so that we have non-empty results we also assume that the topological entropy of this guy is positive. So we want to ask about the number of measures of maximal entropy. So we get one such measure. When does such measure exist? Well a result I think for 98 of Sheldon Newhouse proves that if the map and here the dimension is not relevant, if the map is C infinity then the measure of maximal entropy always exists. So whenever I put a statement or fix a measure of maximal entropy you can think about requiring this extra regularity condition on the map F which is that it is C infinity. So whenever it's C infinity this measure exists. So get one of such measures. I explained to you on the last lecture that whenever the parameter chi that we fix to analyze the quality of hyperbolecy is smaller than the topological entropy well we are actually always assuming that these measures that we fix are at least ergodic. So if you get this measure ergodic of maximal entropy then by the well inequality the measure actually is supported in the non-uniform and hyperbolic locus with this parameter chi. For any chi is smaller than the topological entropy. I think I mentioned this to Lucas on last lecture and I also explained to Viltu two lectures ago because well using well inequality we can get the existence of at least one Lyapunov exponent which is bigger than chi but then if we apply the same inequality for the inverse map because here we are with a different morphism we also get a Lyapunov exponent which is smaller than minus chi so actually the measure lives inside this non-uniform hyperbolic locus with this parameter okay. So you maybe this is this is one of the obstruction to have to extend the results for higher dimension? Not actually well it's one obstruction for the applications because in higher dimension what we do we fix a chi and then we define this subset nuh chi which is the set of points which have Lyapunov x1 bounded from chi in absolute value but if we want to make applications for higher dimensions then measures of a high entropy not necessarily live inside this set. This actually happens not only for the non-invertible situation also for the invertible one okay. Yes okay so if you know that this measure lives in this in this guy then just applying recurrence and Birkhoff's ergodic theorem you can actually conclude that the measure lives in our better set which is the nuh sharp which is actually the set that we are able to construct the Markov partition on. And why is this good? Well because we recall that the the main result at least in this setting due to Sarig is that you can get this topological Markov shift sigma with small sigma which is the left shift and a coding map which restricted to this non-uniformly hyperbolic locus with the sharp you have finiteness 2 1. Having finiteness 2 1 is perfect for lifting measures so if you get your measure of maximum entropy here on M it lives inside this set in which you have finiteness 2 1 then you can lift this measure mu to a measure nu on the symbolic space and because of the finiteness 2 1 this lifted measure actually has the same entropy of the projected one so I could actually put here that you can lift this in a way that the entropy of mu with respect to f is equal to the entropy of mu with respect to sigma okay and in particular this implies to us that this lifted measure is a measure of maximum entropy in the symbolic space. So here we get this because of the finiteness 2 1 you preserve entropy and then the lifted measure is a measure of maximum entropy and why is this good? Well because people came before us and studied exactly the problem of the existence and the number of such measures in this symbolic spaces so this is due to Gurevich in two papers one from 69 and one from 70 in which he considers discountable topological Markov shifts and he proves that they have at most countably many measures of maximum entropy. So what he actually proves is that each measure of maximum entropy ergodic so I should put here ergodic each ergodic measure of maximum entropy lives in a transitive component of the graph so the graph having countably many vertices it has at most countably many transitive components each transitive component supports at most one measure of maximum entropy so automatically you get this result so using this Sarig then obtained the first result on the number of measure of maximum entropy in this context he proved that they are at most countably many of them so the summary of this theorem was leave the measure of maximum entropy to the symbolic space apply Gurevich and then retranslate this result at the symbolic level back to the geometrical model so you get this theorem okay but the question that comes into play now is well how can you go beyond this countability result and prove for example the finiteness of these measures of maximum entropy or even better if you require some weak topological assumptions on the map for example transitivity can you get uniqueness of the measure of maximum entropy this is indeed true in the uniformly hyperbolic context and this is due to Bowen because Bowen understood that when the map F is transitive then the symbolic model that you construct is also transitive so this is also related to a question that Lucas made on the last lecture this result is actually inside the book of Bowen so you can take a look at this classical book of Bowen and exactly because each of these transitive Markov shifts in here recall they are even finite they support exactly one measure of maximum entropy which is the Perry measure so automatically for uniformly hyperbolic defaults either a nozzle or axiom A you get the uniqueness of the measure of maximum entropy it actually exists because this this Markov shifts are finite so they always exist and the unique okay and the question is what happens in this non-informal hyperbolic context well first of all this measure might not even exist so it's not clear that this measure exists why because again the Markov shift that you get from it is countable and it's not true that every countable Markov shift has a measure of maximum entropy okay but if it exists are there finitely many of them well this question was at least to my understanding fully solved by this paper of Buzikro Vizier and Sarik which I will mention as BCS from now on so what does this paper do they use this nice idea that was introduced by Rodrigo's hats Rodrigo's hats Tazibi and Ures of considering homoclinic classes of measures so you know this notion of homoclinic class class is a kind of an old notion in the uniformly hyperbolic context so they introduced it in the in the in the setting of measures exactly to try to relate this notion with the existence of SRB measures SRB measures they are a particular case of equilibrium states they are the measures that are equilibrium states for the geometrical potential minus log of the norm of the derivative in the unstable direction and well you can characterize them as being the measures for which the this integration along unstable manifolds is absolutely continuous with respect to lab bag so they are the kind of substitutes of the lab bag measure in non-conservative situations and the way that they did this study about these SRB measures was basically in two steps they proved and recall here I'm still mentioning everything in low dimension okay they have also some results in higher dimension but for the purpose of this presentation these two steps are focusing only in dimension too so what are the two steps that they use to study these SRB measures they proved that every SRB measure is related to to an homoclinic class in the sense that it lives inside one single homoclinic class so if you know this you are reduced to understanding how many homoclinic classes can carry an SRB measure if you prove that for instance this number is finite then you get the finiteness of SRB measures step two they prove that every homoclinic class indeed can support at most one SRB measure so steps one and steps two reduces the the study on the number of SRB measures to starting how many homoclinic class do have one SRB measure okay and they prove that actually if this map f is topologically transitive then at most one homoclinic class can have SRB measures and because this homoclinic class can have at most one SRB measure automatically that exists at most one SRB measure whenever you have a surface difomophism that is at least of regularity c1 plus beta just like us because they make use of passive invariant manifolds and this map is also topologically transitive okay so this is the kind of sketch of the proof of course there is an extra difficulty here which is that they also prove that if you get two SRB measures which are in the same homoclinic class in some sense in the metric point of view which is the way that this homoclinic class are weakly defined then because you have this nice regularity of the SRB measure so they are actually lab bag measures on the unstable direction you can apply a sards lemma and prove that this kind of weak transversality if it this weak transversality occurs somewhere then it actually you actually have to have actual transversality like differential transversality in another place and this here you are making use exactly of the fact that the unstable the disintegration disintegration in the unstable manifolds is the lab bag measure okay they also make use of low dimensional arguments in which in order to get this transversality they reduce the analysis to some sort of what is now called by buzikro vizier and sarik su rectangles so these are rectangles whose boundary is made of pieces of stable and unstable manifolds of a hyperbolic periodic point so in summary the idea that they make uses sards lemma in order to actually get a real transversality and they use the low dimensionality of the of the model in order to get the transversality so in some sense the result of buzikro vizier and sarik which deals with the non-uniformly hyperbolic context is very inspired in this kind of sketch of proof the difference is that well also in the case of srb measures these homo clinic classes contrary to the uniformly hyperbolic situation that we are used to learn in an initial class of dynamical systems these classes they might not be this joint and you can actually have many of them so this is a bad situation but the good the good point is that well this was proved in in this paper the intersection of these homo clinic classes it carries no entropy okay so this is this is good and then they go further on the analysis and they prove and here is is is very related to exactly what bow and did in the uniform hyperbolic situation they prove that if we reduce yourself to try to understand the dynamics only in a single homo clinic class then the symbolic space that you need is actually a transitive can be taken to be a transitive symbolic space so in some sense this is telling us that inside each homo clinic class because what you get at the symbolic level is transitive then you can have at most one measure of maximal entropy inside each homo clinic class so now the question is to bound the number of homo clinic classes that can carry a measure of maximal entropy so they prove that indeed every every measure with positive entropy lives inside a homo clinic class but the problem is that you could have a bunch of homo clinic classes each of them carrying a measure of maximal entropy so how how do you how do you bound the number of homo clinic classes that carry a measure of maximal entropy well just to mention that these results that they prove here that every measure is supported in a homo clinic class it makes use of a new a more more difficult dynamical sards lemma exactly because you don't know what is the disintegrate in this integration of these measures in the invariant manifolds so in general you have to consider this kind of fractal space that comes up as the intersection of the invariant manifolds this guy you can relate its house of dimension with the regularity of the map i believe they prove that the house of dimension is at least one over r if the map is cr and then using this they they can prove that the these measures with positive entropy in actually entropy bigger than some constant times one over r is supported in this homo clinic class so i mentioned to you that you have these bounds on the house of dimension so what actually is essential here on the contrary contrary to the previous results for the srb measures is that you also need an extra uh much higher regularity on f you need a regularity on f in order to estimate the house of dimensions of these intersections and then apply sards lemma and again as in rodriguez has rodriguez has tazibi uras you need the low dimensionality in order to apply the idea of su rectangles okay the idea of su rectangles allows you to have these su rectangles which separate your manifold into two parts this is no longer true if you go to higher dimensions yes wilton low dimensional what what what do you mean about it this is uh in this stable direction one dimensional and then unstable what do you mean about it it's both so recall that we are considering surface defamofisms okay yeah with positive entropy so each each uh stable and unstable has dimension one okay and that is essential for having these rectangles which separate this space into two parts yeah maybe if you have some quad dimension one uh unstable direction maybe you can do something like that you can separate you can separate if you only use unstable but you'll never be possible because you have like pieces of unstable here but then they stable is just a line something like that so maybe i can add something you read to answer him wilton in fact you have examples in three dimensions so in three dimensions one of these guys is one dimension in partial hyperbolic setting you have examples robust examples of defamofisms in three dimensional manifold in fact in three three torus such that you have more than one measure of maximal entropy in fact like can example you remember can example you have intermingled basins of srb measure you have a two srb measure so here in the in the case of three dimensional partial hyperbolic defamofism you can have transitive transitive defamofisms with two measure of maximal entrant with negative exponents and one bit positive exponents so this is proved in a result by my students juas heli as hosha it will be published in this ice script okay thank you yeah thank you so it's a high dimension new phenomena can appear so recall that we are here in low dimension and our goal is even assuming the highest regularity on f like like it's infinity plus some weak topological property like topological transitivity to to obtain that there is a unique measure of maximal entropy so it's contrary to disconnect samples which you have more than one okay so if if each measure is supported in a homo clinic class if each homo clinic class carries at most one measure of maximal entropy now you are reduced to prove how many homo clinic class have big entropy so buzikoro visie and sarig also prove that like under some regularity assumptions on on f like fc r then there are finitely many of these homo clinic class with large entropy here i put the quoted large because how large exactly depends on how regular your map is but if you reduce ourselves to see infinity maps then this large means just positive entropy so if you if you restate this result of them for c infinity maps then they prove that there are finitely many homo clinic class with positive entropy each of them carrying at most one measure of maximal entropy so automatically if you are c infinity then you get finitely many measures of maximal entropy and if you are transitive and here is the the the result i i believe in the most beautiful uh statement way it's stated at least in my perspective then you prove that again if you are a c infinity surface deep homophism with positive topological entropy that is transitive then there is a unique measure of maximal entropy again the existence is due to new house and their contribution is to prove that this is actually a unique measure of maximal entropy okay okay so this is a kind of a summary of the way that they obtain this result so they they they use many things of this homo clinic class of this previous paper of the four guys rodriguez has rodriguez has tazibi and uris and they introduce new tools in particular this new dynamical sarge lemma together with this understanding that each homo clinic class can be actually coded by a transity mark of shift so this is in the spirit of bowens approach for uniform hyperbole hyperbolecy plus a bunch of new other tools in order to get this result okay so this is all i wanted to mention about measures of maximal entropy for surface defomophisms what can we say on the other context that we have been discussing in this mini course so let me switch the context to the context of three dimensional flows so recall that whenever we consider such flow we require that the vector field defined by it is at least one plus beta and it is non-zero everywhere so it is a vector field without fixed points such just like geodesic flows on surfaces and again for not stating empty results we require the topological entropy of this guy to be positive okay so the way that we apply the how can we understand about this ergodic theoretical properties of measures of maximal entropy in particle so as i told you the way of analyzing flows and this goes back to poincare is constructing poincare return maps so we construct a global poincare section and and we get the like return map which i call poincare map and then we want to relate the non-uniform hyperbolicity of phi with the non-uniform hyperbolicity of f and at least at the level of the derivative and this is all we need in order to define these sets this is okay the only assumption that you need is that the the roof function is bounded away from zero infinity then you can for some chi you can find a chi prime which i believe is like chi times the infimum of this roof function and then you relate these two sets but then if you want to relate to go further because recall we do not code this set directly we call the sharp subset of it but if you want to go further and try to understand the relation actually at the level of the star subset which is requiring recall the star is requiring that limit 1 over n log of the capital Qs along the trajectory is zero and the capital Q it actually depends not only on the derivative but also on how close your trajectory is to the boundary of the section so because of the boundary of the section it is not clear how to relate this set as defined intrinsically for the flow with the respective set defined for the net f so for instance you can ask yourself is this set here non-empty well i know it's non-empty because it carries all periodic trajectories of the flow for example so it's non-empty but from the ergodic theoretical point of view it could be very small it could carry for example no invariant measures with positive entropy exactly because of these boundary effects that points could potentially be approximating the boundary of this section exponentially fast and the consequence of this would be that this set here perhaps would only contain periodic trajectories this would be very bad for our applications you get a measure of maximal entropy for the flow and this measure of maximal entropy would not live inside the set that we are able to code so how do we bypass this difficulty actually what we do is that instead of getting the the actual the original result of SARIG that allows you to code all measures with some Lyapunov exponents bounded away from chi at the same time we fix a measure that is invariant for the flow that has this Lyapunov exponent bigger bigger than chi and then for this particular measure we find a Poincaré section for which this set actually carries the measure and what do I mean by that I mean that with respect to this measure every trajectory of the flow as seen as a trajectory of the return map the Poincaré return map does not converge exponentially fast to the boundary of the section the way we do this is actually constructing a one parameter family of candidates for sections and then by using a double counting argument together with Borel-Cantelli we showed that for most choices of parameters this section has these good properties this this good property that points to not approach the boundary of the section exponentially fast and the conclusion is that the flux measure associated to this measure mu it actually lives inside the non-uniform hyperbolic locus star consequently we can see this measure as living inside the set in which we can make the coding and so the conclusion is that in the above context we can also prove that phi has at most countably many measures of maximal entropy okay you can ask myself can you go beyond and prove that it has finitely many measures of maximal entropy well for that you would have to re-prove all results of Buzikrovis and Sarip for the Poincaré return map the Poincaré return map has singularities and part of the proof of this result of Buzikrovis and Sarip makes use of Fiondin's theory which is only known to hold so far for like smooth maps so it's not clear that you could be able to adapt these results to the Poincaré return map of the flow and then also get finiteness and even uniqueness of this measure of Max Moe entropy so at the flow level this is the best result we have so far okay okay so this treats three-dimensional flows and going further in our list of interesting examples we now discuss billiard maps so recall that billiard maps they are modeled by these maps F defined on surfaces outside what I call the singular set and outside this we assume that this map is well defined and it has this regularity so we assume that this map has topological entropy whatever topological entropy means in this context okay because well you are allowing singularities to occur so uh you have many different definitions of topological entropy it's not clear that they give you a finite number and it's not clear that they are all equal well at least for the class of systems that we are interested that actually for a subclass of them uh balladin dammers were able actually to prove that the right the the usual notion of topological entropy makes sense they are all equal the classical ones and they give you a finite number for example in this context oh I don't have the picture here let me go a little bit down because I know I have it here for billiards like this Lawrence gas topological entropy is a well-defined number it's finite and all usual definitions of topological entropy give you the same number so this is a result of balladin dammers well nevertheless imagine that you have this number which is positive whatever it is and what whatever whatever uh it is defined and now it comes into play a new uh notion which is that of adopted measure recall we always want to say things about measures and now uh we can only say things about measures that are adopted what do I mean by measures that are adopted measures that in some sense do not approach the x the the singular set exponential effects this is basically what this is saying this notion that actually appeared in Sinai's old result because he understood that uh smooth measures are adapted and it also appears a lot in the one-dimensional context okay well uh if you have a measure that is now adapted so points are not converging exponentially fast to the singular set and that are hyperbolic then this measure they live inside the set and inside the set or inside a subset of it we have the coding so we can understand about these measures uh the usual problem is that we use we we need the notion of adaptiveness in these statements because we don't we don't know contrary to the surface differential morphism situation that measures with large entropy are adopted it could occur for an instance that you you had measures of maximal entropy so they are as hyperbolic as you expect but for which uh they are not adapted they are supported towards close very close to the singular set so in general this is a wide open question uh for the bigger situation but gladly something is known so Baladin Damer is also for the class of uh the same class that I mentioned the previous result about the topological entropy for example the Lorentz-Gas they proved that this measure of maximal entropy it actually is unique this is a great result and on top of that it is adapted so being adapted and being a measure of maximal entropy it lives inside the set and then we can understand many things about it for instance we can prove that this measure is Bernoulli well they actually proved this directly using a churn of uh Haskell argument and I would like to mention that this result of Baladin Damer has nothing to do with symbolic dynamics okay they prove it using this uh different machinery of anisotropic spaces but I will I will use these results further in our applications okay so this is uh currently what is known about the subclass of billiard maps in two dimensions and what is known about existence and uniqueness of measures of of max one entropy outside of this class nothing is known and again the techniques used to guarantee this result are no symbolic are not symbolic at all okay so the last setting that I want to mention about measures of maximal entropy is uh actually there are actually two last settings the first one is the higher dimensional version of Saric so you get a c1 plus beta diffeomorphism on a higher dimensional manifold and you assume it has topological entropy so Benovadia uh he proved actually it was again using the coding together with Gurevich's result that f has at most countably many hyperbolic measures of maximal entropy whenever you are in the higher dimensional situation getting a measure of maximal entropy does not guarantee to you that it is hyperbolic it could have some Lyapunov exponents equal to zero imagine for instance and a nozzle of map like the cat map times any rational rotation measures of maximal entropy will have a zero Lyapunov exponent in this circle direction but if you restrict yourself only to the hyperbolic ones then you have at most countably many of them okay finally we arrive at the last setting that I want to mention about measures of maximal entropy the number of measures of maximal entropy and it's that uh more complicated setting as in the result of Araujo Lima and Poletti well I will not mention again that uh that context but it's a context that covers both higher dimensional flows also high dimensional billiards and also non-invertible maps just like the Vienna map okay and the usual problem here is to relate uh large entropy both with hyperbolicity that's the problem that you have for the femorphisms and now with the adaptiveness of the measure so you actually have two things that do not come for free just by getting a measure with large entropy so the result that we have it actually has to restrict ourselves two measures of maximal entropy that are both hyperbolic and adapted and just like Sarig and Benovada we prove that there are at most countably many of them okay basically because these properties guarantee to us that the measure of maximal entropy lives inside the non-uniformly hyperbolic locals with the sharp symbol which is where we get the Markov partition so this concludes the discussion on uh measures of maximal entropy at least how many of them we have in this bunch of examples that we have discussed now we want to say something about their ergodic theoretical properties actually not only of them but more generally of the equilibrium states so you gather an equilibrium state so hold the continuous potential you want their ergodic theoretical structure the good thing is that at a symbolic level if you get such measures then you can apply uh what is the name uh monstein theory and prove that they are actually either Bernoulli or Bernoulli times a rotation so recall that I'm always getting invariant measures that are ergodic so whenever you have ergodic ergodicity the symbolic uh complication given given by your graph guarantees to you well together with some regularity that's given by this whole the continuity of the potential guarantees to you that your measure is either Bernoulli or Bernoulli times a rotational factor so how do we use this in order to get a classification of the measures from the measure theoretical point of view for the geometrical models that we are considering well Sarig used this and proved that if you have this uh equilibrium state of hold the continuous potential if it has positive positive metric entropy and again Sarig's result is on low dimension so that positive metric entropy automatically implies to you hyperbolicity of the measure so it's hyperbolic it lives inside the nuh subset then you can lift it to the symbolic space in the symbolic space the measure is either Bernoulli or Bernoulli times a rotation and this property is preserved under factor maps so the original measure is actually Bernoulli or Bernoulli times a rotation so Sarig obtained this result in this way and in particular it applies to all the measures of maximum entropy that can exist okay great at the flow level we have something similar actually very similar so the measure is either Bernoulli or Bernoulli times a rotation but we have an an extra property that is the following if your flow additionally to have in these properties here that we are always assuming is a contact flow then actually this condition this situation here cannot occur so automatically the measure is actually Bernoulli contact flow is a kind of a quantitative non-integrability assumption on the stable and unstable foliations one example of contact flow is geodesic flows so in particular if you get a measure of maximum entropy of a geodesic flow in a surface with non-positive curvature non-positive curvature implies the flow is no uniform or hyperbolic then this measure of maximum entropy is Bernoulli so this is a consequence of our result okay okay well what can we say on the higher dimensional defilmophism situation again sorry your excuse me about your result with Lima with Sarig you prove for every hyperbolic measure or for just for equilibrium states do you need equilibrium states yeah we use we lift it to the symbolic space and we okay and we use what we know about the equilibrium states of all the continuous potential in the symbolic space okay I understand this let me refine my question so the last part when you are assume contact the structure I mean assume that you have the you know that it is Bernoulli or Bernoulli times rotation then I mean just this to this to to discard the Bernoulli time rotation you need just contact destruction and some invariant measure which is hyperbolic no do you need at this level that yeah yeah yeah yeah yeah yeah actually we just need to know that the measure is mixed if we know it's mixing then the Bernoulli times rotation is discarded exactly so so this was my quote yeah yeah and the contact the contact structure guarantees to you that in some sense the measure is mixing this is actually at least at the level of geodesic flows this is due to Babio Babio she she she showed she she gave an argument to go from the ergodicity using hops a kind of hops argument to go actually from our ergodicity to mixing okay thanks okay so for higher dimension as I told you you always have to assume on top of having a positive topological entropy you actually have to assume the measure is hyperbolic so the same result of study holds here so this measure is either Bernoulli or Bernoulli times of rotation and finally we have the result on this more complicated setting of Araujo Lima and Paletti in which on top of assuming that the measure is an ergodic equilibrium state of a whole the continuous potential with positive entropy it is hyperbolic and adapted and if it satisfies these two properties then the measure is either Bernoulli or Bernoulli times of rotation of course if you are in the non-invertible context actually we are not saying this about the measure but of its counterpart in the natural extension because we have to Bernoulli is only a property that's well defined for invertible maps so we go to the natural extension and then the natural extension of this measure is either Bernoulli or Bernoulli times of rotation okay so here unfortunately we do not know if this contact version for three dimensional flows holds for high dimensional ones so it's an open question to know that if the flow is contact to prove that the measure is Bernoulli to discard the Bernoulli times rotation case in some cases this is known actually for like geodesic flows in rank one manifold in the same context of this paper of Bunts, Klimerhaga, Fischen and Thompson. Cole and Thompson proved that these in equilibrium states they have the K property so having the K property you can discard automatically the Bernoulli times rotation situation and get as a consequence of our result that it is actually Bernoulli okay okay so this finishes the discussion that I wanted to make about measures of maximal entropy I now want to go to this second topic about the number of periodic trajectories so I just introduced some notation these four maps this guy counts the number of periodic points with period N and for flows this guy counts the number of closed trajectories with period at most N and to obtain information about these numbers we should first obtain information about these numbers at the symbolic level well this was satisfactorily understood by Gurevich in the same two papers so he proved that for countable topological marker of shifts with positive topological entropy if the guy has a measure of maximal entropy which is not always true then the number of periodic points is at least this guy so this proves that the periodic points grow at least exponentially fast at a rate given by the topological entropy okay so the idea is to use this result of Gurevich to get the same result at the geometrical model this is what Sarik did so he considered the same setting as usual and if there is a measure of maximal entropy for example I already mentioned to you that if the map is infinity this measure exists then you can get the counting of the periodic points actually what Sarik did was getting a count of periodic points only for multiples of some number p so this number p that appears here is a kind of period possible period that appears in the symbolic space in the topological mark of shift so the original work was not able even assuming transitivity of the diffeomorphism to prove that the symbolic space that you get or you can redo the symbolic space in order to get a kind of topologically mixing symbolic space and so getting this p equal to 1 so he was only able to get this counting along multiples of a constant p okay gladly Buzi was actually able to kind of get the symbolic coding given by Sarik and do a new symbolic coding using the idea of magic words that he introduced to prove that if you actually have a transitive you can redo the coding to get a transitive coding and so get the counting of periodic points for every n not only along multiples of n okay this was an improvement to get rid of the period the possible period that might appear here in the symbolic space well benovadia has the same result so if you get if your map has a hyperbolic measure of maximum entropy then there is a p for which this guy these numbers you have an estimate only along multiples of p and again you can apply Buzi and if additionally f is transitive then you can get rid of the p and prove that this estimate estimate holds for every mode every iterate of f okay so this deals with the diffeomorphism situation what can we get about the billiard situation well we are always assuming the existence of measures of maximum entropy so we have to reduce ourselves to the context of Baladin and Amherst so if we for example consider this billiard map then using the coding given by myself and Matheus together with the existence and adaptiveness of the measure of maximum entropy together with the getting rid of period result of Buzi we can prove that the number of periodic points of period n is also has this estimate here okay this is nice because this uh it was known by Stoyanov both in dimension two in higher dimensions that these numbers for billiards they can grow at most exponentially fast so we proved that they actually grow exponentially fast and at in some sense the best rate that we can can get which is that of the topological measure okay so this is the result for diffeomorphisms and for billiards and what is the result for flows for flows the right result is a kind of Margulis estimate which you have estimates of the order of constants times e to the ht divided by t and this is exactly what myself and Sarigi proved Sarigi proved again assuming the existence of a measure of maximum entropy because it's only under this assumption that we have good average so if this occurs then we have a Margulis like estimate for the number of periodic closer trajectories of land at most t okay finally uh the result in this higher generality of Araujo Lima and Poletti uh on top of requiring the existence of a measure of maximum entropy requires that this measure is also hyperbolic if you have a flow again for flows okay so I was mentioning here flow so I'll take the chance to mention the result of Araujo Lima and Poletti for flows we have a flow with positive topological entropy that has a measure of maximum entropy that's hyperbolic then again we can apply that one parameter trick of Poincaré sections to construct a section for which this measure is also adapted and so it lives in the non-uniformly hyperbolic locals in which we construct the Markov partition having done that we can lift the measure to the symbolic space count the periodic trajectories at the symbolic space getting some estimate of this order and then concluding that the same estimate holds for the original flow so this is the sketch of the proof of this result well still in the context of Araujo Lima and Poletti we mentioned flows what can we say about non-invertible maps and about billiards well for non-invertible maps the the one we are mostly interested in for application are the Vienna maps the non-invertible version of the Vienna maps and we proved that the number of periodic points is at least again a constant times e to the topological entropy times n okay okay finally we have the result for billiards so for billiards we restrict our attention for those billiards that we know that the srb that the natural smooth invariant measure of these guys is non-uniformly hyperbolic so for example for this three-dimensional version of the Lorentz gas or for this three-dimensional version of the Bunimovich stadium so for these guys it is known that the srb natural smooth invariant measure is hyperbolic and it has positive metric entropy which i call h so applying the result for this h which by previous people is hyperbolic and because it is the srb measure it is a smooth measure it is also adapted then we can get that the number of periodic points again satisfies this okay so i believe i am done so this is the last result that i wanted to mention so i fulfilled my promise of speaking only for one hour today tonight you can make questions or comments on whatever you want Yuri in this result about the number of periodic points for Vienna maps you have a a low bounded about it but this the limit that is you have a upper bound of the number of periodic points no that's converged to okay no uh i actually don't know how to do it i mean recall that we are only coding the part of the dynamics that has some non-uniform hyperbolicity so if you have periodic points that are not hyperbolic we are not able to see them okay okay but you don't have a way to relate the the number of periodic points in this context there's in this particular context of Vienna maps with entropy of this part of the phase space that is not expanding something like that uh yes then these are exactly the periodic points that we count but the the result of Gurevich it only guarantees that you have this lower bound so even at the symbolic level we cannot get a asymptotic estimate on this so Gurevich shows that there is a vertex v0 such that the periodic points on the symbolic space with v0 equal to vn equals to this v sorry is of the order of each of the h times n so at least the periodic points grow at this rate but he does not prove that these the periodic points of the symbolic of the the shift is exactly of this order okay okay so this is a this is a kind of uh uh even if you restrict yourself for the hyperbolic periodic trajectories it is a thing that from the symbolic point of view we still do not fully understand okay I'd like to mention that Vadim Kaloshin has examples of difiumophisms where the number of periodic orbits goes super expansion and hyperbolic yeah okay this example is uh in two-dimensional what what are kind of I don't remember I don't remember thank you I I think it is he he he he gets he does things near two homo clinic tangences okay well one thing also of these Kaloshin's results is that I believe that the multiplier of the periodic trajectories goes to zero so you could still ask yourself what happens for the periodic trajectories with multiplier at least chi right okay Ali and Omri did I did I say many wrong things about your works no I don't remember my works very well no but I think that was okay thank you for explaining it to you yes so this is a a good place to stop because you see next week we're going to have the workshop right we're gonna have two lectures each day and Omri Jehom and Silvan they they are actually going to make three lectures all correlated to each other exactly mentioning this result that I didn't mention today about the exponential decay of correlations so everybody is invited to attend the workshop next week the the program is already at the website would you like to ask a question thank you sorry I just wanted if possible would you explain a bit about the results in the non-invertible case also which parts are done or or you think it can be done in such cases so in the non-invertible case recall that we only we do not code the map itself but we code its natural extension okay so we have yeah exactly so you have that projection var theta that I introduced so we code this guy so every ergodic theoretical property of this guy that can be translated to this guy we are able to understand for example periodic points periodic points here are the same as the periodic points here so that's how we get the result for Vienna maps counting periodic points in the natural extension okay for invariant measures if you get an invariant measure on n then you have its lift to m hat for instance if you want to say something about the Bernoulli property for meal you cannot say it directly because it's only defined for invertible systems so you say something about meal hat so its version in the natural extension the unique lift of it in the natural extension will be either Bernoulli or Bernoulli times of rotation then you can retranslate what Bernoulli here means below I think it's uh exact I don't recall now what is the non-invertible version of Bernoulli the Bernoulli property I don't know as well maybe yeah I don't know something like just once I did Bernoulli measures I didn't understand the question because in your case you were talking about some kind of symbolic Bernoulli measure because you fix some code and say that thing that called the the measure is Bernoulli and uh but uh okay well we actually proved that this measure in the natural extension which is fully given if you give me the measure I know what measure meal hat is in the natural extension that is only one yes but in the uh in the natural extension you're using the code that you present and to see if the measure is Bernoulli or not yes but it's intrinsic the Bernoulli here of meal hat is intrinsic of the measure recall that I have an extension here for which if I lift this measure let me use blue meal hat to the symbolic space then I prove that it is Bernoulli or Bernoulli times of rotation here so it is Bernoulli here because or Bernoulli times of rotation but because this property is preserved on the projection on their factors it will also be here but they can only go up to here because going down I will go to a possible non-invertible map for which the Bernoulli property is not well defined and I think that at the non-invertible level it means that meal is exact yes I understand but uh your symbol in the top uh live over here your yes your say you have this uh coding that you present depends on some parameters like uh the expansion to to make this so if uh you you going to say that if you have a measure that can be lifted to this coding so you can say that in this code the the measure is Bernoulli or not yes and then it and then the original one is yes but uh yeah this this is not a uh uh how do I say uh canonical code on something like that that you can say uh there are universal coding no something like that because depends on your parameter of expansion it's something like that yes actually the construction depends on many parameters yes so this is uh uh uh Bernoulli relative to some coding yes but for that is enough to prove that the original one is Bernoulli right but Bernoulli in which sense because uh in the measure theoretical sense that this guy it is measured theoretically isomorphic to a full shift with a Bernoulli measure you have a by measurable uh all invertible almost everywhere map for which the there is a Bernoulli measure for which this by measurable map is sending this measure to okay so as I told you Bernoulli here means measured theoretically Bernoulli so Markov measures for us are actually Bernoulli because they are measured theoretically isomorphic to Bernoulli measures okay Yuri something that I commented with you maybe I take advantage here it is related to this non-invertible case it is an old question that I had I have written in text so if you have a non-invertible map so consider for instance a 3111 on the two torres 3111 then it is degree two map and you may hope that it is uh coded by some one sided shift which which is 221 but it is not because its entropy is larger than logarithm of two is 3111 I think this this was uh announced by Hank Ruin this this example well this is a simple example that shows that you may have some non-invertible map degree two but it is not coded by uh one sided shift of 221 of course maybe it can be coded with some 321 as a sub shift of a 321 or 421 but as I can see you prove that okay it's it's it's natural extension is Bernoulli so you have some two sided shift but as I can understand still we don't know if there are some one sided model for for endomorphisms yes yes yeah it's still not known in some sense you were telling me the answer right you were telling me that uh even in the expanding situation yes you should consider the stable stable direction right not necessarily even if you want a complete one sided shift yes I think but if you maybe you can embed it into some other higher dimensional in some 321 421 one sided shift I mean four symbols symbols more than two symbols for these guys yeah what what our result gives is a invertible symbolic extension right from here to here of the review I think no maybe maybe I don't know if you are if you want ergodic properties I don't know if there is something that we need to go really to one sided shift but in in terms of ergodic theoretic properties probably you have done everything no yeah well if you go to more intrinsic properties like isomorphism isomorphic to something like yes structure then perhaps you should restrict yourself to the class that you start with but for the application that we have in mind this is not okay and as I mentioned you before if you have some hyperbole uniform hyperbole case you answered me that yes this shift will be finitely many symbols will have finitely many symbols yes and actually if you are if you use I think Bowen's approach if you are the uniformly expanding situation then you can get a one sided shift okay but mm-hmm okay I will continue the question so this macro partition on the natural extension will induce a finite to one relation with the map on M or not no this guy is highly non-finite yeah yeah okay so just we have the information up to uh inverse limit space inverse limit map in fact yeah thank you it's highly non-finite one because at each pre iteration you have finitely many choices of branches okay yes thank you but the beauty of the natural extension is on this result of rockling that measures on it are related to measures below and vice versa yeah even though this is infinite to one for instance uh in case of uh measure of maximum entropy and equilibrium states is there any results uh relate yes so so exactly if you if you get a measure of maximum entropy here it will be a measure of maximum entropy here this is how we prove our result if it's in the uh equilibrium state of a whole the continuous potential here it will be an equilibrium state of how the whole the continuous potential here yeah very nice thank you thank you poya Yuri yes I have a simple question which is a concept that I don't know what's it uh so you have said that this measure new in the symbolic space in some situations it would be uh Bernoulli or Bernoulli times rotation what is the rotation measure in the symbolic space so it's uh I mentioned to you being Bernoulli it's like uh being a measure that's measured theoretical isomorphic to a full shift with a Bernoulli measure mm-hmm so it's like zero one okay k minus one to the z and then you consider a rotation in a finite set zero one p minus one the map that sends uh zero to one one to two p minus one p minus one to zero so you have a product map here here you have the shift here you have this plus one map okay so here you have Bernoulli measures here you have the the uniform measure so Bernoulli times rotation that means exactly this this product measure on this product space it is as if you have many shifts in different levels and the dynamics is actually sending this to here this to here this to here in this to here in a way that if you do it here four times then you get something that uh is actually like a shift so if you if you have ever seen the spectral deposition of a uniformly hyperbolic the thermophism this is the this is the possible presence of the period inside the the basic sets okay okay i got it so the measure new which is Bernoulli times rotation is actually not defined in the in the shift space it's defined in this product space that you have written here well it will be defined in this shift space but you have a major theoretical isomorphism between this shift space and the product of a full shift space with this finite set in which this measure theoretical isomorphism sends the measure new to this Bernoulli measure times this uniform measure in this finite set okay thanks thanks i have a question about the lower bounds on the number of periodic orbits for billiard tables because if i am not wrong the there were results for dispersive billiard tables already in the ninths so how does the results you you said are more general or the bounds are better or there are more tables for so i i think what chernoff proved in 91 was that limit 1 over n log of the periodic points is greater than or equal to this h so in particular you have this result if instead of h you have h minus epsilon for any epsilon okay so in some sense we get rid of the h and here the h is exactly the entropy of the srb measure because it's the only one that we know it's hyperbolic and it has positive entropy if for some reason like baladi dammers we prove that there are measures with higher entropy and hyperbolic then this h can we will be able to be improved automatically not directly because we will not know that the measure is adapted but if we prove that the measure is also adapted then yeah we can improve this estimate the other result from the 90s that it was known about this is the one dimension of stoionov that proves that these guys they are they grow at the most exponentially fast okay so in the high dimensional situation uh it wasn't i i believe that chernoff's result actually it's only restricted to two dimension two if i'm not mistaken this result so i think the only result i remember i checked the literature carefully the only result i found was stoionov that gave an upper bound thank you that is one extra thing i wanted to say is that in the non-reform expanding situation vilton has a different construction for like getting symbolic extensions of these guys right as far as i understand vilton uses inducing skin yes uh so i can do a propaganda so in my talk i'm going to talk about if in particular i pretend to show that it's a joint work with paul varandas to show that for vienna maps you have only a measure of maximal entropy and all potential with small variation hold the potential also have a unique equilibrium states for vienna maps interesting something like that and this will be with this kind of code in that uh and indeed it's uh we can construct some kind of in full induced map that every measure with uh in in this context so expand the measure with the high entropy is lift liftable to this inducing map in the for instance the context of vienna maps so so in this case you can apply for instance sarige and gurevich uh fear to prove the unity of uh equilibrium states and something like that so it's in this direction very nice i think this result was announced before right about the uniqueness of measure yes but uh we spend a lot of time trying to uh give a for for a measure of maximum entropy the all the calculation was done five or six or more years ago but uh we are trying to to do a uh independent proof because uh i'm going to talk about this but yeah we say ah forget about this let's do the uh using uh many other stuff and uh and conclude the results so now the uh the problem with uh uh you use the gurevich pressure is that we don't know what what uh okay you can show that you have only one equilibrium state for for instance the gurevich pressure is finite it's bounded for upper have an upper bound but you don't know if this imply at least i don't know if this imply uh that uh this the this equilibrium states is uh equilibrium states in the the closer of the lift ball measure for this in this map so for for uh maximum entropy you can do that and we are trying to do in general and but uh for now it's not was very hard uh we do that for some kind of potential not only the zero potential but in general not so we are bored for it for for this time this okay but the results we are going to see next week yeah yeah it actually reminds me of asking you title and abstract no but i i i i i send i may for you as the and the straffano with the abstract and also i put it in the platform of the ctp and so yeah it was automatically asked by stefano yeah so but okay good yuri thank you now for your beautiful talks well yeah i think you should thank you it was really wonderful very illuminating so yeah exactly thank you thank you guys for standing me for more than nine hours thank you see you later so no other questions so thank you all again for being here and tomorrow we'll continue with the mini course of z the third lecture the mini course of z let me recall you again that uh next week as as we were advertising here we have the workshop okay we always occur at the same time slot from um at well at least here in brazil is from 10 to 12 10 to 12 30 actually we have slightly changed so see you all maybe tomorrow maybe next week bye bye see you